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Volume densities with the mean value property for harmonic functions. (English) Zbl 0818.31003
Let \(U\) be a bounded domain in \(\mathbb{R}^ d\) \((d\geq 2)\) such that \(0\in U\), and let \(\lambda\) denote Lebesgue measure on \(U\). It is shown that there exists a (strictly) positive \(C^ \infty\) function \(w\) on \(U\) such that \(h(0)= \int hw d\lambda\) for every \(h\) in \({\mathcal H}_ b (U)\), the set of bounded harmonic functions on \(U\). Further, if \(\partial U\) is of class \(C^{1+ \varepsilon}\), then it can be arranged that \(\inf w(U)>0\). The smoothness condition on \(\partial U\) cannot be omitted: the authors construct a domain \(U\) such that if \(w\) is a nonnegative measurable function on \(U\) which satisfies \(h(0)= \int hw d\lambda\) for every \(h\) in \({\mathcal H}_ b (U)\), then \(\inf w(U)= 0\). This answers a question posed by A. Cornea at the International Conference on Potential Theory in Nagoya in 1990.

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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